GREKing 100 Must Solve Quantitative Comparison Questions

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GREKing 100 Must Solve Quantitative Comparison Questions 1. Quantity A Quantity B the number of even integers the number of even integers between 1 and 13 between 2 and 14

2. Quantity A Quantity B the area of ΔQ 8 sq m 3. Quantity A Quantity B √0.16 √0.0016 4. Quantity A Quantity B the average (arithmetic the average (arithmetic mean) of 6, 5, 8, 7, and 9 mean) of 11, 2, and 8 5. Amy, Megan, and Sharon divided a batch of cookies among themselves. Amy took 30% of the cookies and Sharon took 40% of the cookies. Amy ate 1/3 of the cookies she took and Sharon ate 1/4 of the cookies she took. Quantity A Quantity B number of cookies Amy ate number of cookies Sharon ate



6. Quantity A Quantity B the slope of the line in 3/2 the graph 7. Quantity A Quantity B p − 8 p + 8 8. Quantity A Quantity B the average (arithmetic mean) the average (arithmetic of 16, 23, 30, 45, and 17 mean) of 23, 18, 17, 35, and 45 9. Quantity A Quantity B 31x 35x

10. Quantity A Quantity B the area of the circle in the area of a circle with the figure diameter 3y 11. a < 0 Quantity A Quantity B a2 a3



12. A = {15, 20, 20, 13} Quantity A Quantity B the median of set A the mode of set 13. w < x < y < z Quantity A Quantity B wx yz 14.

x ≥ 3 Quantity A Quantity B the volume of the box the volume of the cylinder on the left on the right 15. Quantity A Quantity B the percent increase from the percent increase from 10 cm to 14 cm 54 cm to 58 cm 16. The mean of set B is 17. B = −5, −1, 12, 29, x, y Quantity A Quantity B X y 17. Quantity A Quantity B 1/25 0.016/4



18. Quantity A Quantity B X z 19. Quantity A Quantity B √8 + √13 √19 + √9 20. C = 3, 6, 11, 12, 10, 18, x The mean of set C is 9. Quantity A Quantity B 8 x 21. Quantity A Quantity B 5 + √32 √23 + 4

22. Quantity A Quantity B 9 x 23. Marvin sells candy bars at a rate of 3 bars for $4. Quantity A Quantity B at this rate, the cost of x dollars x candy bars



Use the following sequence to answer questions 24 and 25. .25, .5, .75, 1, 1.25, 1.5, 1.75, 2, . . . 24. Quantity A Quantity B the 53rd term of the sequence 13 25. Quantity A Quantity B the 78th term of the sequence 19.5 26. Quantity A Quantity B (x + 2)2 (x − 2)2

27. Quantity A Quantity B 6b the sum of the interior angles of the polygon above 28. Quantity A Quantity B 8.7 × 368 9 × 368 29. Quantity A Quantity B 18 – 4/5 + 1/2 18 – 1/2 + 4/5 Use the following series to answer questions 30 and 31. 2 + 4 + 6 + 8 + . . . + 98 + 100 30. Quantity A Quantity B the sum of the first 23 550 terms of the series 31. Quantity A Quantity B the sum of all the terms 2,550 in the series



32. Quantity A Quantity B the area of the largest circle the area of the largest circle that that can be cut out of a can be cut out of a rectangular square piece of paper with piece of paper with length of sides of 3.4ʺ 3.2ʺ and a width of 5ʺ 33. Quantity A Quantity B 1/20 of 800 5% of 800 34. n < 0 Quantity A Quantity B 7n 4n

35. Quantity A Quantity B AC 4 36. Quantity A Quantity B p2 −p2 37. Quantity A Quantity B (85 − 93)(22 − 8) (42 − 95)(11 − 17) 38. ΔQRS is an isosceles triangle with angles Q = 45° and R = 45° and line segments QS = 8 and QR = x. Polygon DEFGH has sides DE = 3 and GH = y and polygon LMNOP has sides LM = 1 and OP = 2. Quantity A Quantity B x y 39. x ≥ 0 Quantity A Quantity B x2 x3 40. Quantity A Quantity B 5 × (3 + 1) ÷ 2 5 × 3 + 1 ÷ 2



Use the following figure to answer questions from 41 to 45

41. Quantity A Quantity B the measure of ∠AOD the measure of any right angle 42. Quantity A Quantity B the measure of ∠AOB the measure of an acute angle 43. Quantity A Quantity B the measure of a reflex angle the measure of ∠FOA 44. Quantity A Quantity B the measure of an angle the measure of an angle supplementary to ∠BOC supplementary to ∠FOE 45. Quantity A Quantity B the sum of the measures of the sum of the measures of the ∠AOF, ∠AOD, and ∠BOD interior angles in a square 46. There are 150 people in a movie theater. 75 of the people are men, 60 are women, and the remainder are children. Quantity A Quantity B percent of people in the 10% theater that are children 47. Quantity A Quantity B 0.75% 0.0075 48. The ratio of dogs to cats in a pet store is 5:3. There are 96 dogs and cats in the store. Quantity A Quantity B the number of dogs 65 in the pet store



49. Quantity A Quantity B 45% of 104 43 50. Quantity A Quantity B 117 + 23 25 Use the following figure to answer questions from 51 to 58

51. Quantity A Quantity B the measure of ∠1 the measure of ∠3 52. Quantity A Quantity B the measure of ∠1 the measure of ∠5 53. Quantity A Quantity B the measure of ∠7 the measure of ∠3 54. Quantity A Quantity B the sum of the measures the sum of the measures of of angles 5 and 8 angles 2 and 3 55. Quantity A Quantity B the measure of ∠2 the measure of ∠8 56. Quantity A Quantity B the measure of ∠3 the measure of ∠6 57. The measure of ∠1 is 100°. Quantity A Quantity B 75° the measure of ∠8



58. The measure of ∠3 is 105°. Quantity A Quantity B the measure of ∠6 77° 59. a is a positive integer. Quantity A Quantity B 1/a7 1/a10 60. Quantity A Quantity B (a4)2 a8 61. Quantity A Quantity B 33 9 62. a is an integer. Quantity A Quantity B a2a3 a6 63. Quantity A Quantity B (3/4)2-(1/4)2 (1/2)2 64. 4x = 64 Quantity A Quantity B x 4 65. 212 = 8x Quantity A Quantity B 3 x 66. a < 0 Quantity A Quantity B a2 a9 67. y = 5x x is a positive integer. Quantity A Quantity B 5x + 1 5y 68. x > 0 Quantity A Quantity B (2x + 4)(x + 1) 2x2 + 5x + 4 69. x2 – 4x – 21 = 0 Quantity A Quantity B sum of the roots product of the roots




70. The measure of ∠3 is 100°. Quantity A Quantity B the measure of ∠4 the measure of ∠8 71. The measure of ∠2 is 65°. Quantity A Quantity B 110° the measure of ∠11 72. The measure of ∠9 is 95°. Quantity A Quantity B the measure of ∠16 the measure of ∠8 73. The measure of ∠1 is x. Quantity A Quantity B the measure of ∠8 2x 74. The sum of the measures of ∠13 and ∠10 is 160°. Quantity A Quantity B the measure of ∠4 the measure of ∠11 75. The prom committee orders an arch for the entrance to the dance floor. The arch follows the equation y = 2x – 0.1x2 where y is the height of the arch, in feet. Quantity A Quantity B maximum arch height 10 feet 76. x < 0 Quantity A Quantity B x(x + 7) x2 + 7



77. x > 0 Quantity A Quantity B (x + 3)2 x2 + 9 78. y = x2 + 6x + 9 Quantity A Quantity B minimum y value 9 of the function Use the following figure to answer questions from 79 to 80

79. Quantity A Quantity B the number of sides the number of sides of this polygon of a triangle 80. Quantity A Quantity B the sum of the interior 360 angles of this polygon 81. y = 4x2 + 4x – 8 Quantity A Quantity B smaller root −2 82. y = −x2 + 6x Quantity A Quantity B larger root 6




83. Quantity A Quantity B the sum of the interior the sum of the exterior angles of this polygon angles of this polygon 84. Quantity A Quantity B the number of sides the number of sides in in this polygon a heptagon 85. Quantity A Quantity B the sum of the interior the sum of the interior angles of this polygon angles of a hexagon 86. 9x2 = 6x – 1 Quantity A Quantity B x 1 87. Quantity A Quantity B (x + 5)(x – 5) x2 + 10x – 25 88. xy < 0 Quantity A Quantity B (x + y)2 x2 + y2 89. 2(x + 3) + 6 = 4x Quantity A Quantity B 6 x 90. Julie is 5 years older than Ravi. Three years ago, Julie was twice as old as Ravi Quantity A Quantity B Ravi’s age now 5




91. Quantity A Quantity B the sum of the interior the sum of the interior angles angles of an 8-sided polygon of this figure 92. Quantity A Quantity B 1/2 of the sum of the interior the sum of the interior angles angles of this figure of a triangle 93. Quantity A Quantity B the area of this polygon the area of a convex polygon if all sides have a length of 8 whose interior angles measure 900. 94. The sum of 3 consecutive integers is 37 more than the largest integer. Quantity A Quantity B the middle integer 19 95. A convex polygon has 5 sides. This polygon also has three right angles and two congruent angles. Quantity A Quantity B the measure of one of 130° the congruent angles 96. The sale price for a snowboard is $63.00. This price reflects a 30% discount. Quantity A Quantity B the original price $90.00 of the snowboard



97. Quantity A Quantity B number of freshmen 22 98. There are 400 students enrolled at Brown High School. Quantity A Quantity B number of seniors 120 99. Quantity A Quantity B difference between the difference between the quantity of freshmen quantity of juniors and sophomores and seniors 100. a, b, and c are integers greater than 1 and (cb)5 = c5 + a Quantity A Quantity B a b



Answers 1. a. There are 6 even integers between 1 and 13. They are 2, 4, 6, 8, 10, and 12. There are 5 even integers between 2 and 14. They are 4, 6, 8, 10, and 12. ` 2. b. The area of a triangle = (1/2)bh. ΔQ is a right triangle, so we can substitute 3 and 4 for b and h. A = (1/2)(3(4)) = (1/2)(12) = 6. The area of ΔQ is 6 sq m, which is less than 8 sq m, so the answer is b. 3. a. The square root of 0.16 is 0.4. The square root of 0.0016 is 0.04. 4. c. The mean of the numbers in column A is 7: 6 + 5 + 8 + 7 + 9 = 35. 35 divided by 5 = 7. The mean of the numbers in column B is also 7: 11 + 2 + 8 = 21. 21 divided by 3 = 7. 5. c. Amy took 30% of the cookies and ate 1/3 of those, which is 10% of the original number of cookies. Sharon took 40% of the cookies and ate ¼ of those, which is also 10% of the original number of cookies. Since both women ate 10% of the original number of cookies, they ate the same amount. 6. c. The slope of a line is defined as !"#$%&($)

!"#$%&($*. The line in the graph crosses the origin at

(0,0) and also intersects the point (2,3). This creates a rise (change in the y-value) of 3 and a run (change in the x-value) of 2, which gives a slope of 3/2. Thus, the values in a and b are equal, so the answer is c. 7. b. 8 more than any number (p + 8) is more than 8 less than that number (p – 8). 8. b. Of the 5 numbers in each set, 3 are the same (23, 17, and 45). The remaining 2 numbers are greater in the set in column B than in the set in column A, therefore, the mean of the set in column B is greater than the mean of the set in column A. 9. d. The relationship cannot be determined. If the value of x is 0, then both quantities are 0. If the value of x is positive, then 35x is greater than 31x. If the value of x is negative, then 31x is greater than 35x. 10. d. The area of a circle is defined as A = πr2. The radius of the circle drawn is 2x, so we can calculate the area of the circle as 4x2π. The area of a circle with diameter 3y would be a = πr2 = (3y)2π = 9y2π. However, since the values of x and y are not defined, it is impossible to evaluate whether quantity A (4x2π) or quantity B (9y2π) is greater. If, for instance, x = 5 and y = 2, then quantity A would be greater. But if x = 2 and y = 5, quantity B would be greater. So there is not enough information to evaluate the equations and the answer is d. 11. a. Since a is a negative number (a < 0), a2 is a positive number because a negative times a negative is a positive; a3 is a negative number because three negatives multiply to a



negative answer. A positive is always greater than a negative, so quantity A is greater than quantity B. 12. b. The median of set A (the middle of the set when the numbers are put in numerical order) is 17.5: 13, 15, 20, 20. Since there is an even number of elements in the set, the average of the two middle numbers, 15 and 20 is found; 15 + 20 = 35; 35/2 = 17.5. The mode of set A (the most frequently occurring element in the set) is 20 because it occurs twice. 20 is greater than 17.5. 13. d. The relationship cannot be determined. If w = −10, x = −9, y = 1, and z = 2, then wx = 90 and yz = 2, so quantity A is greater. If w = 0, x = 1, y = 2, and z = 3, then wx = 0 and yz = 6, so quantity B is greater. Either quantity can be greater, depending on the choice of variables. 14. b. The formula for the volume of a box is A = lwh. So the volume of the box at left = 4(6)x or 24x. The volume of a cylinder = πr2h, so the volume of the cylinder at right = πx2x, or πx3. Substituting x = 3 into both equations, the volume of the box becomes 24(3) = 72 and the volume of the cylinder becomes π(3)3 = π(27) = 3.14(27) = 85.78. So long as x is greater than or equal to 3, the volume of the cylinder is greater and the answer is b. 15. a. Both quantities increased by 4, but quantity A increased 4 from 10, or 40%, and quantity B increased 4 from 54, or 7.4%. 40% is greater than 7.4%. 16. d. The relationship cannot be determined. Since the mean of set B is 17, and the set has 6 elements, the sum of the elements in the set must be (17)(6) = 102. The total of the given elements is −5 + −1 + 12 + 29 = 35. Therefore, x and y must total what’s left, namely, 102 − 35 = 67. There is no way to know which of the two might be greater, or if the two are equal. 17. a. Change 1/25 to a decimal by dividing 1 by 25 to get .04. Change 0.016/4 to a decimal by dividing .016 by 4 to get .004. .04 is greater than .004. 18. d. When two parallel lines are cut by a line segment, the resulting corresponding angles created are equal, so x = y. The sum of two complementary angles is always 180, so x + z = 180, and since x = y, it is also true that y + z = 180. However, no information is given about the relationship between x and z other than the fact that they add up to 180. Even though the figure makes it look like x > z is true, this cannot be taken for fact. Therefore there is not enough information to further evaluate the problem and the answer is d. 19. b. Compare √8 to √9 and find that √9 is greater and comes from quantity B. Compare √13 to √19 and find that √19 is greater and comes from quantity B. Since both parts of quantity B are greater than the parts of quantity A, quantity B is greater.



20. a. Since the mean of set C is 9 and the set has 7 elements, the sum of the elements in the set must be (9)(7) = 63. The total of the given elements is 3 + 6 + 11 + 12 + 10 + 18 = 60. Therefore, x must equal the remaining value, namely, 63 − 60 = 3. 8 is greater than 3. 21. a. Compare 5 to 4 and find that 5 is greater and comes from quantity A. Compare √32 to √23 and find that √32 is greater and comes from quantity A. Since both parts of quantity A are greater than the parts of quantity B, quantity A is greater. 22. a. The Pythagorean theorem states that in any right triangle, a2 + b2 = c2 where c is the hypotenuse of the triangle and a and b are the lengths of the other sides. Since the side labeled 10 is opposite the right angle, it is the hypotenuse of the triangle, so the Pythagorean theorem can be used by substituting 6, x, and 10 for a, b, and c, respectively. A2 + b2 = c2 62 + x2 = 102 36 + x2 = 100 x2 = 64 x = 8 x = 8 and therefore is less than 9, so the answer is a. Note: If you recognize that 6, 8, 10 is a Pythagorean triple, then you know that x must be equal to 8 and you can quickly solve the problem. 23. a. Each candy bar costs more than $1 (divide $4 by 3). Therefore, the cost of x candy bars is more than x dollars. 24. a. The pattern in this series (.25, .5, .75, 1, 1.25, 1.5 . . .) can also be written as 1/4, 2/4, 3/4, 4/4, 5/4, 6/4 . . . . Since each term given is equivalent to the term number (its number in the sequence) divided by 4, an nth term would be equal to n 4 . The 53rd term is equal to 53/4; 53/4 = 13.25; 13.25 > 13, so the answer is a. 25. c. Use the same equation you established in the previous problem. The 82nd term = 82/4 = 19.5. Column B is also 19.5, so the correct answer is c. 26. d. The relationship cannot be determined. If 0 is substituted for x, both quantities are 4. If 6 is substituted for x, quantity A is 64 [(6 + 2)2 = 64] and quantity B is 16 [(6 – 2)2 = 16]. In the first example, the quantities are equal and in the second example, quantity A is greater. The relationship cannot be determined. 27. b. When two line segments intersect, the resulting vertical angles are always equal, so b = 80°. Therefore, 6b = 6(80°) = 480°. The sum of the interior angles of a polygon can be found by drawing all diagonals of the polygon from one vertex and multiplying the number of triangles formed by 180°. The polygon at the right can be divided into 3 triangles, so 3(180°) = 540°. 540° > 480°, so the answer is b.



28. b. Both quantities contain the number 368. In quantity B, 368 is being multiplied by a larger number, making quantity B greater than quantity A. 29. b. Consider there to be an equal sign between the two columns. Subtract 18 from both sides, leaving −4/5 + ½ = −1/2 + 4/5. Since 4/5 is larger than ½, the left side of the equation (quantity A) is negative and the right side (quantity B) is positive. A positive number is always greater than a negative number, so quantity B is greater. 30. a. Note that the first term is 2, the second term is 4, the third term is 6, etc. The term is equal to two times the term number. Therefore, the 23rd term is 23 × 2 = 46 and the sum of the even numbers from 2 to 46 is needed to answer the question. In series questions, an easy shortcut is to rewrite the series below the original series and add vertically to get consistent sums. In this case, 2 + 4 + 6 + . . . + 44 + 46 46 + 44 + 42 + . . . + 4 + 2 48 + 48 + 48 + . . . + 48 + 48 Since 23 terms were added, there are 23 totals of 48. Each term has been written twice, so 23 × 48 is double the total needed and must be divided by two; 23 × (48/2) = 23 × 24 = 552; 552 > 550, so column A is greater. 31. c. Following the same procedure as in the previous question, the last term is 100, so there are 100/2, or 50 terms in the series. The consistent sum would be 102 because the first term plus the last term = 2 + 100 = 102. There would be 50 sums of 102 which would again be double the total needed; 50 × (102/2) = 50 × 51 = 2,550. 32. a. The area of a circle is dependent on the length of its radius, so the problem here is to determine which circle could have the largest radius. To cut a circle out of a rectangular piece of paper, you must draw a circle whose radius is no greater than any of the sides of the rectangle. This is because the radius extends equally from the center of the circle in all directions. So even though the width of the rectangle in column B is 5ʺ, its length is only 3.2ʺ, which is smaller than the sides of the square in column A (3.4ʺ). Thus a circle with a larger radius—and, therefore, greater area—could be cut from the square in column A. So the answer is a. 33. c. 1/20 and 5% are the same thing. Therefore, the two quantities are equal. 34. b. n is a negative number. Try a couple of negative numbers to see the pattern. Substitute in −2; (7)(−2) = −14 and (4)(−2) = −8, quantity B is greater. Substitute in −0.5; (7)(−0.5) = −3.5 and (4)(−0.5) = −2. Again, quantity B is greater. 35. c. Sketching ΔABC will be helpful here. ∠B must be right because the sum of angles A and C is 90°, and the sum of all three angles in the triangle must add up to 180°. The fact that this triangle has angles of 30°, 60° and 90° means that it is a 30-60-90 special right



triangle whose sides are x, x√3 and 2x. (You should have the lengths of this and the other special right triangle—a 45-45-90 triangle—memorized.) Since AC is the hypotenuse of this triangle, its value can be represented by 2x according to the relative lengths of the sides of this kind of special right triangle. BC measures 2√3, and it is opposite the 60° angle (∠A), so it can be represented as the x√3 side of the triangle. Setting 2√3 = x√3, and dividing both sides of the equation by √3 yields x = 2. Plugging this value of x into AC = 2x gives line segment AC a value of 2(2), which equals 4. Thus the values of column A and column B are the same, and the answer is c. 36. a. Any number squared is positive. Therefore, p2 is positive and (−p)2 is negative. A positive number is always greater than a negative number. Quantity A is greater. 37. b. Both sets of parentheses in quantity B are negative. Two negatives multiplied yield a positive answer. Quantity A has one negative set of parentheses and one positive. A negative multiplied by a positive yields a negative answer. Since quantity B is always positive and quantity A is always negative, quantity A is greater. 38. d. Though the problem tempts you to sketch the shapes and use your knowledge of isosceles triangles to determine x and your knowledge of similar polygons to determine y, there is no need. The problem does not state that polygons DEFGH and LMNOP are similar, and since there is no information indicating that their corresponding sides are in the same ratio or that corresponding angles are equal, this cannot be determined. There is not enough information to solve the problem, so the answer is d. When taking the test, be sure to read through the problems before spending time sketching shapes and solving equations. Determine whether or not you have enough information to solve the problem before delving into it. Note: If you were told that the polygons were, in fact, similar, the problem could be solved. Line segment QR in ΔQRS is the hypotenuse (by virtue of being opposite of ∠s, which must equal 90°) and therefore has a length greater than 8. The fact that corresponding sides in similar polygons have lengths of a similar ratio could then be used to set up the ratio DE/GH = LM/OP. Substituting would yield 3/y = ½, which solves to y = 6. Since x > 8, x must be greater than y and therefore the answer would be a. 39. d. The relationship cannot be determined. If x < 1, for example 0.5, then quantity A is larger. If x = 1, then both quantities are equal (both 1). If x > 1, for example 3, then quantity B is larger. 40. b. Use the order of operations to simplify. 5 × (3 + 1) ÷ 2 = 5 × 4 ÷ 2 = 20 ÷ 2 = 10 = quantity A 5 × 3 + 1 ÷ 2 = 15 + .5 = 15.5 = quantity B 41. a. The measure of ∠AOD as indicated in the diagram is 100°. The measure of any right angle is always 90°. Therefore ∠AOD is bigger and the answer is a.



42. d. The measure of ∠AOB as indicated in the diagram is 20°. An acute angle is an angle with a measure less than 90°. Since an acute angle could be less than 20°, equal to 20° or between 20° and 90°, there is not enough information to say whether or not an acute angle would be greater than ∠AOB. Therefore, the answer is d. 43. a. A reflex angle is defined as an angle whose measure is between 180° and 360°. The measure of ∠FOA as indicated in the diagram is 180° (which makes it a straight angle). This is smaller than a reflex angle, so the answer is a. 44. b. Supplementary angles are angles whose measure adds up to 180°. The diagram does not directly indicate the measures of angles BOC and FOE, but it does give enough information to find these measures using subtraction: ∠BOC = ∠AOC −∠AOB ∠BOC = 70° − 20° ∠BOC = 50° ∠FOE = ∠FOA −∠EOA ∠FOE = 180° − 135° ∠FOE = 45° ∠BOC = 50°, so its supplement must equal 180° − 50°, or 130°. ∠FOE = 45°, so its supplement must equal 180° − 45°, or 135°. The supplement to ∠FOE is larger, so the answer is b. 45. c. The interior angles in a square are all right angles, and since there are four of them, the sum of their measures is always 360°. ∠AOF as indicated in the diagram measures 180° and ∠AOD measures 100°. To find the measure of ∠BOD, use subtraction: ∠BOD = ∠AOD −∠AOB ∠BOD = 100° − 20° ∠BOD = 80°. Therefore the measures of the three angles in column A are 180°, 100°, and 80°, which add up to 360°. This is the same as the sum in column B, so the answer is c. 46. c. There are 15 children out of 150 people; 15/150 = 0.10 = 10%. The percentage of people in the theater that are children is 10%. 47. c. Change .75% to a decimal by moving the decimal point two places to the left. .75% = 0.0075. 48. b. Use the equation 5x + 3x = 96 and solve for x. 5x + 3x = 96 8x = 96 8x/8 = 96/8 x = 12 5x of the animals are dogs. Since x = 12, 5x = 60. There are 60 dogs, which is less than quantity B. 49. a. Notice that 45% of 104 would be more than 45% of 100. Since 45% of 100 is 45, quantity A is greater than 45 which is greater than quantity B. 50. b. The number 1 to any power is 1; 23 is 2 × 2 × 2 which is 8; 1 + 8 = 9. The quantity 25, in column B, is greater.



51. c. Vertical angles are congruent (equal). Vertical angles are defined as angles, formed by 2 intersecting lines, which are directly across or opposite from each other. Angles 1 and 3 are vertical and therefore congruent. The answer is c. 52. c. When a transversal line intersects two parallel lines, the resulting corresponding angles are congruent. Corresponding angles are defined as the angles on the same side of the transversal and either both above or below the parallel lines. Angles 1 and 5 are corresponding and therefore congruent, so the answer is c. 53. c. Angles 7 and 3 are corresponding and therefore congruent. The answer is c. 54. c. Angles 5 and 8 are supplementary because they combine to form a straight line. The same is true of angles 2 and 3. Supplementary angles always add up to 180°, so the measures of both sets of angles are the same and the answer is c. 55. c. Angles 2 and 8 are neither corresponding nor vertical. However, angles 2 and 6 are corresponding, so their measures are equal. Angles 6 and 8 are vertical, so their measures are also equal. This information can be used to determine that angles 2 and 8 are congruent because m∠2 = m∠6 = m∠8. In fact, angles 2 and 8 are called alternate exterior angles. Alternate exterior angles are always congruent, so the answer is c. 56. d Angles 3 and 6 are same side interior angles. This means that they are both inside the parallel lines and on the same side of the transversal. Same side interior angles are always supplementary, so their measures add up to 180°. However, this relationship says nothing about the specific values of each angle, and even though the drawing makes it look like one angle might be larger than the other, no information is given that could determine the actual value of either angle. Therefore, there is not enough information to solve the problem and the answer is d. 57. b. Angles 1 and 8 are same side exterior angles. This means that they are both outside the parallel lines and on the same side of the transversal. Same side exterior angles are always supplementary. Supplementary angles add up to 180° and the measure of ∠1 is given as 100°, so the measure of ∠8 must be 180° − 100° = 80°. 80° is greater than 75°, so the answer is b. 58. b. Angles 3 and 6 are same side interior angles, which means that they are supplementary. Since the measure of ∠3 is 105°, the measure of ∠6 must be 180° − 105° = 75°; 77° > 75°, so column B is greater. The answer is b. 59. a. Since the variable a is a positive integer, both choices are positive, and a10 > a7. These are fractions with the same numerator. When two fractions are being compared with the same numerators, the smaller the denominator, the larger the number. Column A is greater. 60. c. By the laws of exponents, (a4)2 = a4 × 2 = a8. This is true for any real number a. Therefore, the quantities in Column A and Column B are equal.



61. a. 33 is equal to 3 × 3 × 3, which is 27. 27 is greater than 9, so column A is greater. 62. d. The answer cannot be determined. This problem involves a law of exponents that is true for any real number: a2 × a3 = a2+3 = a5. For most integers, a6 > a5. Note that this is true even for negative integers, since 6 is an even number, and 5 is an odd number. There are two exceptions, however, that would make these choices equal. They are when a = 0 or a=1. 63. a. When a fraction is squared, you square both the numerator, and the denominator, so (3/4 )2 = 32/42 = 9/16 and (1/4)2 = 1/16 and 9/16 – 1/16 = 8/16 = 1/2 in lowest terms. Likewise, (1/2)2 = 1/4. One half is greater than one fourth. 64. b. Four to the x power means that 4 is multiplied by itself “x” times. By trial and error, 4 × 4 × 4 = 64, so x is equal to 3. Choice B is greater. 65. b. For exponential equations, you must first rewrite the equation to have the same bases when possible. Since 2 × 2 × 2 = 8, 23 = 8. Two is the common base. Rewrite the equation as 212 = (23)x. By the laws of exponents, (23)x = 23x. The equation is now 212 = 23x. Since the bases are now the same, this becomes a simple equation to solve, by setting the exponents equal to each other: 12 = 3x. Divide both sides of this equation by 3, and it becomes x = 4. 66. a. When a is less than zero, a is negative. A negative number to any even power is a positive number, while a negative number to any odd power is a negative number. This is a case where even though the exponent of 2 is smaller, the quantity will be greater. 67. c. Since y = 5x, multiply each side of this equation by 5, to get 5y = (51)(5x). (51)(5x) =5x+1, which is the value of column A. 68. a. Using the distributive property, (2x + 4)(x + 1) = 2x2 + 2x + 4x + 4. Combine like terms, and this is equal to 2x2 + 6x + 4. Subtracting 2x2 and 4 from both columns leaves 6x in column A and 5x in column B. Since x > 0, column A is greater. 69. a. To find the roots of the equation, factor the left hand side into two binomials; x2 − 4x − 21 = (x − 7)(x + 3), so the equation becomes (x − 7)(x + 3) = 0. Either x − 7 = 0 or x + 3 = 0 to make the equation true. So x = 7 or x = −3. The sum of the roots is 7 + −3 = 4. The product of the roots is (−7)(3) = −21. 70. b. ∠3 and ∠4 are supplementary angles, so the sum of their measures must add up to 180°. Therefore the measure of ∠4 = 180° − 100° = 80°; ∠3 and ∠8 are vertical angles and therefore congruent, so the measure of ∠8 = 100°; 100° > 80°, so the measure of ∠8 is larger than the measure of ∠4 and the answer is b. 71. b. ∠2 and ∠11 have no direct relationship upon first glance. However, there are two sets of parallel lines in this diagram (lines a and b, and lines c and d) so there are many related



angles to work with. ∠2 and ∠3 are same side exterior angles along transversal c and therefore are supplementary. So the measure of ∠3 is 180° − 65° = 115°. Angles 3 and 11 are corresponding angles (along transversal b and both above the parallel lines c and d) and so are congruent. So the measure of ∠11 is also 115°, which is greater than column A. The answer is b. 72. c. The measure of ∠9 is information you actually don’t need to solve this problem. Some questions will provide extra information like this in an attempt to throw you off, so don’t be tricked if you’re sure of how to solve the problem! In this case, angles 16 and 8 are corresponding angles (both along transversal b and beneath parallel lines c and d, respectively) and so must be congruent. Therefore, the answer is c because their measures are the same. 73. b. ∠1 and ∠8 are alternate exterior angles (on opposite sides of transversal c and on the outside of the parallel lines a and b), and therefore are congruent. So the measure of ∠8 is x; x indicates the measure of an angle, so it cannot be negative. Therefore, 2x > x and the answer is b. 74. b. This problem is actually not that difficult to solve, but it does require several steps to determine the measures of both angles since there is no direct relationship between the two. First, angles 13 and 10 are vertical angles and therefore congruent. So the measure of each angle is 1/2 of their sum, or 80°. Next, angles 10 and 11 are same side interior angles (along transversal d and both on the inside of parallel lines a and b) and therefore supplementary, so the measure of ∠11 must be 180° − 80° = 100°. Now find the measure of ∠4. ∠11 and ∠7 are same side interior angles (along transversal b and both on the inside of parallel lines c and d) and so are supplementary. Therefore the measure of ∠7 = 180° − 100° = 80°; ∠7 and ∠4 are vertical angles, and so the measure of ∠4 must also be 80. The measure of ∠11 is greater than the measure of ∠4 because 100° > 80°. The answer is b. With some practice, you will become familiar with the relationships between the angles created by a transversal cutting two parallel lines. This will make analyzing a system of multiple parallel lines and transversals much easier, and you will be able to quickly intuit the relationship between angles like these by logically connecting different pairs of congruent and supplementary angles. Once you know the basic rules it becomes easier and easier to break down the components of a complicated system of lines and angles to solve the problem. 75. c. The arch is in the shape of a parabola, and the maximum arch height (the y value) is the height at the vertex. When a quadratic is in the form ax2 + bx + c (a, b, c are real numbers), the x-coordinate of the vertex is given by the formula -b/2a = -2/-0.2 = 10. When x is 10, y = 2(10) − 0.1(10)2. So y = 20 − 0.1(100) = 20 − 10 = 10. So the maximum arch height is 10 feet. 76. b. By the distributive property, x(x + 7) = x2 + 7x for column A. Since x < 0, x is negative, and therefore 7x is negative, x2 + 7 will be greater in this case.



77. a. The quantity (x + 3)2 = (x + 3)(x + 3). By the distributive property, this equals x2 + 3x + 3x + 9 = x2 + 6x + 9. You can subtract x2 and 9 from each column and you are left with 6x in column A and zero in column B. Since x > 0, column A is greater. 78. b. The given equation is a quadratic in the form ax2 + bx + c, where a, b, and c are real numbers. The minimum y value of the function is the y value at the vertex. The x-coordinate of the vertex is given by the formula -b/2a = -6/2 = −3. When x = −3, y = (−3)2 + 6(−3) + 9 = 9 − 18 + 9 = 0. Column B is greater. 79. a. This polygon has four sides, making it a quadrilateral. Triangles are three-sided polygons. A quadrilateral has more sides than a triangle, so the answer is a. 80. c. You can use the formula S = 180(n − 2) to find the sum of the angles in a convex polygon, where n represents the number of sides in the polygon. A four-sided polygon such as this one has an angle sum of S = 180(4 − 2) = 180(2) = 360, which is equal to the amount in column A, so the answer is c. 81. c. To find the roots of this equation, first factor out the common factor of 4; 4x2 + 4x − 8 = 4(x2 + x − 2). Next, factor this into two binomials; 4(x2 + x − 2) = 4(x + 2)(x − 1). The roots are the values of x that make the y value equal to zero. The equation will equal zero when x = −2 or x = 1. The smaller root is −2. 82. c. To find the roots of this equation, factor out the common factor of −x. The equation becomes y = −x(x − 6). The roots are the values of x that make the y value equal to zero. The equation will equal zero when x = 0 or x = 6. The larger root is 6. 83. a. You can use the formula S = 180 (n − 2) to determine that the angle sum of this polygon is 900. The exterior sum of any convex polygon is always 360, so the answer is a. 84. c. A heptagon is a seven-sided polygon. This polygon also has seven sides, so the values in the two choices are equal. The answer is c. 85. a. Though the formula S = 180 (n − 2) can be used to determine that the angle sum of this polygon is 900 and the angle sum of a hexagon is 720, an understanding of the nature of convex polygons provides an easier way to solve the problem. A polygon’s angle sum increases as the number of sides of the polygon increases. Since this polygon has more sides (7) than a hexagon (6), the sum of its interior angle sum will be greater, so the answer is a. 86. b. To find the values of x, move all terms to the left side so that it is a quadratic equation set equal to zero. Subtracting 6x and −1 from both sides makes the equation 9x2 − 6x + 1 = 0. Factor this quadratic into two binomials; 9x2 − 6x + 1 = (3x − 1)(3x − 1). Now the equation is (3x − 1)(3x − 1) = 0. This will be true when 3x − 1 = 0; add one to both sides to give 3x = 1; divide both sides by 3 and x = 1/3.



87. d. The answer cannot be determined. The binomials in column A are the difference of two squares, so (x + 5)(x − 5) = x2 − 5x + 5x − 25 = x2 − 25. Since there is no indication as to whether x is positive or negative, the term 10x in column B could be either positive or negative and the answer cannot be determined. 88. b. Using the distributive property and combining like terms, column A is (x + y)2 = x2 + xy + xy + y2 = x2 + 2xy + y2. Since xy < 0, 2xy is negative, and thus column B is greater. 89. c. To solve this equation, first use the distributive property, and combine like terms, on the left hand side. The left side becomes 2(x + 3) + 6 = 2x + 6 + 6 = 2x + 12. Now, 2x + 12 = 4x. Subtract 2x from each side, so 12 = 2x. Divide both sides by 2, and x = 6. 90. a. Set up an equation and let x represent Ravi’s age now. Julie’s age now is represented by x + 5. Three years ago, Ravi’s age was x − 3 and Julie’s age was x + 5 − 3 = x + 2. Three years ago, Julie was twice as old as Ravi. So x + 2 = 2(x − 3). Use the distributive property on the right hand side to get x + 2 = 2x − 6. Add six to both sides, yielding x + 8 = 2x. Subtract x from both sides, and x = 8. Ravi’s age now is 8. 91. c. This figure is an 8-sided polygon, so the value of choices are equal. The answer is c. 92. a. Though you can use the formula S = 180 (n − 2) to determine that 1/2 of the angle sum of this polygon is 540 (the entire sum is 1,080) and the angle sum of a triangle is 180, you should more quickly be able to determine that since this is an 8-sided polygon, its angle sum will be more than double that of a 3-sided triangle. In either case, the answer is a. 93. d. This figure is an 8-sided polygon with all sides of length 8. Though the math is a bit involved, you do have enough information to determine its area. However, while the sum of the interior angles of a convex polygon can be used to determine how many sides the figure has, the area of the second polygon cannot be calculated without more information. Therefore this problem cannot be fully solved and the answer is d. 94. c. Set up an equation where x represents the first consecutive integer. Therefore the second consecutive integer is x + 1, and the third is x + 2. The sum of these integers is represented by x + x + 1 + x + 2. Combining like terms, the sum is represented by 3x + 3. This sum is 37 more than the largest integer, so 3x + 3 = x + 2 + 37. Combining like terms on the right hand side yields the equation 3x + 3 = x + 39. Subtracting x from both sides yields 2x + 3 = 39. Subtract 3 from both sides gives 2x = 36. Divide both sides by 2, and x = 18. Now, x represents the smallest integer, so the middle integer is x + 1 = 19. 95. a. The formula S = 180(n − 2) can be used to determine that the sum of the interior angles of this polygon is 540. Right angles measure 90° each, so subtracting the sum of the three right angles from 540° leaves 270° for the two remaining congruent angles. Congruent angles are angles with equal measures, so dividing (270/2)° yields 135° for each angle, which is larger than the value in column B (130°). The answer, therefore, is a.



96. c. $63.00, which is the sale price, is 70% of the original price. Therefore, let x represent the original price of the snowboard. So 0.70x = 63.00. Divide both sides of this equation by 0.70, and x = $90.00. 97. d. The relationship cannot be determined. Since the number of students enrolled at Brown High School is not given for this question, it is not known whether the number of freshmen equals 22 (if there are 100 students, because 22% of 100 = 22), is more than 22 (if there are > 100 students, because 22% of > 100 is > 22) or is less than 22 (if there are < 100 students, because 22% of < 100 is < 22) 98. b. Since there are 400 students at Brown High School and 29% are seniors, 29% of 400, or .29 × 400 = 116; 120 > 116, so the answer is b. 99. b. The difference between the number of freshmen and sophomores is 24% − 22% = 2% of 400; .02 × 400 = 8 students. The difference between the number of juniors and seniors is 29% − 25% = 4%. 4% of 400 is 16. 16 > 8. 100. a. Simplify the exponents on the left-hand side of the equation by multiplying. Then, since the bases are the same (c), the exponents can be set equal to each other: (cb)5 = c5 + a c5b = c5 + a 5b = 5 + a Isolate a by subtracting 5 on both sides of the equation. 5b − 5 = 5 + a − 5 5b − 5 = a The variables must be integers greater than 1 (so the smallest possible value is 2). When 2 is substituted in for b, a is equal to 5. As the value of b gets larger, so does the value of a. Quantity A is always greater than quantity B.

GREKing 100 Must Solve Quantitative Comparison Questions

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